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Representation of j=1 rotation matrix

  1. Apr 12, 2008 #1
    [SOLVED] Representation of j=1 rotation matrix

    The derivation of this involves the use of the following fact for j=1:

    [atex]\frac{J_y}{\hbar} = (J_y/\hbar)^3[/itex].

    Is there a simple way to see this other than slogging through the algebra by expanding out the RHS using [itex]J_y = \frac{1}{2i}(J_+ - J_i)[/itex] and [itex]J_{\pm}|jm\rangle = \hbar\sqrt{(j\mp m)(j \pm m + 1)}| j,m\pm 1\rangle[/itex]?
     
  2. jcsd
  3. Apr 12, 2008 #2

    pam

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    That's not true in general, since the eigenvalues of J_y for j=1 are +1,0,-1.
    It is not true for j=-1. Perhaps there is a special circumstance in your problem.
     
  4. Apr 12, 2008 #3
    Hi Pam,

    It is true for j=1, and I was being stupid anyway, the trick is to use matrix multiplication.

    Ie just write down [itex]\langle j'=1,m'| J_y|j=1,m\rangle[/itex] and do the trivial matrix multiplication. Doh!
     
  5. Apr 12, 2008 #4

    pam

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    Of course, I made the silly mistake of thinking (-1)^3=+1.
    Using the evs is a simpler proof than even trivial matrix math.
     
  6. Apr 12, 2008 #5
    evs?
     
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